What is the general solution for cos^2(x)=(1-cos(2x))/2 ?

The general solution for cos^2(x)=(1-cos(2x))/2 is x=(3pi)/4+pin,x= pi/4+pin

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cos^2(x)=(1-cos(2x))/2

Solution

cos^{2} (x) = \frac{1 - cos ( 2 x )}{2}

Solution

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

Degrees

x = 13 5^{\circ} + 18 0^{\circ} n, x = 4 5^{\circ} + 18 0^{\circ} n

Solution steps

cos^{2} (x) = \frac{1 - cos ( 2 x )}{2}

Subtract

\frac{1 - cos ( 2 x )}{2}

from both sides

cos^{2} (x) - \frac{1 - cos ( 2 x )}{2} = 0

Simplify

cos^{2} (x) - \frac{1 - cos ( 2 x )}{2} : \frac{2 cos ^{2} ( x ) - 1 + cos ( 2 x )}{2}

cos^{2} (x) - \frac{1 - cos ( 2 x )}{2}

Convert element to fraction:

cos^{2} (x) = \frac{cos ^{2} ( x ) 2}{2}

= \frac{cos ^{2} ( x ) \cdot 2}{2} - \frac{1 - cos ( 2 x )}{2}

Since the denominators are equal, combine the fractions:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{cos ^{2} ( x ) \cdot 2 - ( 1 - cos ( 2 x ) )}{2}

Expand

cos^{2} (x) \cdot 2 - (1 - cos (2 x)) : cos^{2} (x) \cdot 2 - 1 + cos (2 x)

cos^{2} (x) \cdot 2 - (1 - cos (2 x))

= 2 cos^{2} (x) - (1 - cos (2 x))

- (1 - cos (2 x)) : - 1 + cos (2 x)

- (1 - cos (2 x))

Distribute parentheses

= - (1) - (- cos (2 x))

Apply minus-plus rules

- (- a) = a, - (a) = - a

= - 1 + cos (2 x)

= cos^{2} (x) \cdot 2 - 1 + cos (2 x)

= \frac{2 cos ^{2} ( x ) - 1 + cos ( 2 x )}{2}

\frac{2 cos ^{2} ( x ) - 1 + cos ( 2 x )}{2} = 0

\frac{f ( x )}{g ( x )} = 0 \Rightarrow f (x) = 0

2 cos^{2} (x) - 1 + cos (2 x) = 0

Rewrite using trig identities

- 1 + cos (2 x) + 2 cos^{2} (x)

Use the Double Angle identity:

cos (2 x) = 1 - 2 sin^{2} (x)

= - 1 + 1 - 2 sin^{2} (x) + 2 cos^{2} (x)

Simplify

= 2 cos^{2} (x) - 2 sin^{2} (x)

2 cos^{2} (x) - 2 sin^{2} (x) = 0

Factor

2 cos^{2} (x) - 2 sin^{2} (x) : 2 (cos (x) + sin (x)) (cos (x) - sin (x))

2 cos^{2} (x) - 2 sin^{2} (x)

Factor out common term

2

= 2 (cos^{2} (x) - sin^{2} (x))

Factor

cos^{2} (x) - sin^{2} (x) : (cos (x) + sin (x)) (cos (x) - sin (x))

cos^{2} (x) - sin^{2} (x)

Apply Difference of Two Squares Formula:

x^{2} - y^{2} = (x + y) (x - y)

cos^{2} (x) - sin^{2} (x) = (cos (x) + sin (x)) (cos (x) - sin (x))

= (cos (x) + sin (x)) (cos (x) - sin (x))

= 2 (cos (x) + sin (x)) (cos (x) - sin (x))

2 (cos (x) + sin (x)) (cos (x) - sin (x)) = 0

Solving each part separately

cos (x) + sin (x) = 0 or cos (x) - sin (x) = 0

cos (x) + sin (x) = 0 : x = \frac{3 π}{4} + πn

cos (x) + sin (x) = 0

Rewrite using trig identities

cos (x) + sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) + sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 + \frac{sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 + tan (x) = 0

1 + tan (x) = 0

Move

1

to the right side

1 + tan (x) = 0

Subtract

1

from both sides

1 + tan (x) - 1 = 0 - 1

Simplify

tan (x) = - 1

tan (x) = - 1

General solutions for

tan (x) = - 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{3 π}{4} + πn

x = \frac{3 π}{4} + πn

cos (x) - sin (x) = 0 : x = \frac{π}{4} + πn

cos (x) - sin (x) = 0

Rewrite using trig identities

cos (x) - sin (x) = 0

Divide both sides by

cos (x), cos (x) \neq = 0

\frac{cos ( x ) - sin ( x )}{cos ( x )} = \frac{0}{cos ( x )}

Simplify

1 - \frac{sin ( x )}{cos ( x )} = 0

Use the basic trigonometric identity:

\frac{sin ( x )}{cos ( x )} = tan (x)

1 - tan (x) = 0

1 - tan (x) = 0

Move

1

to the right side

1 - tan (x) = 0

Subtract

1

from both sides

1 - tan (x) - 1 = 0 - 1

Simplify

- tan (x) = - 1

- tan (x) = - 1

Divide both sides by

- 1

- tan (x) = - 1

Divide both sides by

- 1

\frac{- tan ( x )}{- 1} = \frac{- 1}{- 1}

Simplify

tan (x) = 1

tan (x) = 1

General solutions for

tan (x) = 1

tan (x)

periodicity table with

πn

cycle:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} tan (x) 0 \frac{3}{3} 13 \pm \infty - 3 - 1 - \frac{3}{3}

x = \frac{π}{4} + πn

x = \frac{π}{4} + πn

Combine all the solutions

x = \frac{3 π}{4} + πn, x = \frac{π}{4} + πn

Popular Examples

cos^2(x)-4cos(x)-1=0

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-9sin(3x)=0

- 9 sin (3 x) = 0

cos(4x)-7cos(2x)=8

cos (4 x) - 7 cos (2 x) = 8

6tan(B)+sqrt(14)=0

6 tan (B) + 14 = 0

-4tan^2(θ)-4tan(θ)-9=8tan(θ)-4

- 4 tan^{2} (θ) - 4 tan (θ) - 9 = 8 tan (θ) - 4

Frequently Asked Questions (FAQ)

What is the general solution for cos^2(x)=(1-cos(2x))/2 ?
The general solution for cos^2(x)=(1-cos(2x))/2 is x=(3pi)/4+pin,x= pi/4+pin